Comparisons of Turbulence Stresses from Experiments against the Attached Eddy Hypothesis in Boundary Layers

Turbulence stress statistics in a boundary layer at Reτ ≈ 10,000 are measured using custom × hot-wire probes. The results show logarithmic behaviour in the profiles of the streamwise and spanwise turbulence intensities against wall-normal distance in the same region in which the mean velocity exhibits logarithmic behaviour, consistent with the predictions of the attached eddy hypothesis. Comparisons are drawn with computations applying the attached eddy hypothesis using two different typical representative eddies: (i) hierarchies of individual hairpins and (ii) hierarchies of packets of hairpins. Promising results are obtained when a packet-eddy is used rather than an individual hairpineddy, when compared with experimental results in the logarithmic region. Introduction In the attached eddy hypothesis, the turbulent boundary layer is idealised as a collection of randomly arranged geometrically similar representative eddies [11]. From this model, Townsend [11] concludes that at sufficiently high Re, the turbulence intensities follow u2 U2 τ = −A1 ln z δ +B1, (1) v2 U2 τ = −A2 ln z δ +B2, (2) and w2 U2 τ = B3, (3) in the logarithmic region, assuming uw/U2 τ = −1, where Uτ corresponds to the friction velocity, while A1, A2, B1, B2 and B3 are constants which depends on the form of the attached eddies. Throughout the paper, we use the co-ordinate system x, y and z to refer to the streamwise, spanwise and wall-normal directions; with u, v and w denoting the corresponding fluctuating velocities, respectively. Here we use the superscript ‘+’ to indicate viscous scaled quantities, while capitalisation and overbars indicate time averaged quantities. The representative eddies have characteristic heights ranging from δ1, the smallest eddy, to ∆E , the largest eddy of the order of the boundary layer thickness (δ). Eddies of identical characteristic height are referred to as a ‘hierarchy’, with multiple hierarchies representing various energetic scales present in the flow. Figures 1(a) and (b) show two examples of simple representative eddies (further details of the geometries are provided in a later section), while figures 1(c) and (d) show the idealised boundary layers with three hierarchies of representative eddy (i.e. δ1 and ∆E correspond to the characteristic heights of the first and third hierarchy, respectively). It should be noted that the representative eddy is a statistical concept which captures the bulk features of the average eddy shape. In reality, the shape and size of eddies evolve over their life span and hence it is highly unlikely (a) Hairpin-eddy